Exact Stratification and Affine Mass Formulas for Split Richelot Data over Finite Fields

Abstract

The Richelot $(2,2)$-step is the standard step of explicit genus-2 isogeny computation. We determine the exact stratification of its input space over a finite field $\mathbb{F}_q$ of odd characteristic: ordered factorizations $f=uvw$ of a square-free sextic into monic quadratics fall into three strata by the geometric type of the quotient, governed by the incidence geometry of the discriminant locus. This yields closed formulas for each stratum and, modulo affine coordinate changes, mass formulas of degree four in $q$ with a complete classification of stabilisers. The classification is decided by data the step already computes, at $5\mathrm{M}+6\mathrm{S}$ beyond the brackets, and an output post-check is provably redundant. On the decomposable stratum the square class of one resultant determines the field of definition of the elliptic factors and the shape of the Weil polynomial of the Jacobian; the two cases are counted exactly, and in the nonsplit case the curve $y^2=f$ has $q+1$ rational points. Exhaustive enumeration over small finite fields verifies every proved count.